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Hessian Matrices

We are about to look at a method of finding extreme values for multivariable functions. We will first need to define what is known as the Hessian Matrix (sometimes simply referred to as just the "Hessian") of a multivariable function.

Recall from the Clairaut's Theorem on Higher Order Partial Derivatives page that if the second mixed partial derivatives of $f$ are continuous on some neighbourhood of $f$, then these mixed partial derivatives are equal on this neighbourhood. That is for $\mathbf{x} = (x_1, x_2, ..., x_n)$ and $z = f(x_1, x_2, ..., x_n)$ we have that $f_{ij} (\mathbf{x}) = f_{ji} (\mathbf{x})$ for $i = 1, 2, ...,n$ and $j = 1, 2, ..., n$. So the continuity of all of the second mixed partial derivatives of $f$ imply that the Hessian $H(\mathbf{x})$ is symmetric.

Example 1

Find the Hessian Matrix of the function $f(x, y) = x^2y + xy^3$.

We need to first find the first partial derivatives of $f$. We have that: